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Let A be a collection of n linear hyperplanes in k^l, where k is analgebraically closed field. The Orlik-Terao algebra of A is the subalgebra R(A)of the rational functions generated by reciprocals of linear forms vanishing onhyperplanes of A. It determines an irreducible subvariety of projective space.We show that a flat X of A is modular if and only if R(A) is a split extensionof the Orlik-Terao algebra of the subarrangement A_X. This provides anotherrefinement of Stanley's modular factorization theorem and a newcharacterization of modularity, similar in spirit to the modular fibrationtheorem of Paris.  We deduce that if A is supersolvable, then its Orlik-Terao algebra is Koszul.In certain cases, the algebra is also a complete intersection, and wecharacterize when this happens.

Modular decomposition of the Orlik-Terao algebra of a hyperplane arrangement